## You are here

Homeuniform continuity

## Primary tabs

# uniform continuity

In this entry, we extend the usual definition of a uniformly continuous function between metric spaces to arbitrary uniform spaces.

Let $(X,\mathcal{U}),(Y,\mathcal{V})$ be uniform spaces (the second component is the uniformity on the first component). A function $f:X\to Y$ is said to be *uniformly continuous* if for any $V\in\mathcal{V}$ there is a $U\in\mathcal{U}$ such that for all $x\in X$, $U[x]\subseteq f^{{-1}}(V[f(x)])$.

Sometimes it is useful to use an alternative but equivalent version of uniform continuity of a function:

###### Proposition 1.

Suppose $f:X\to Y$ is a function and $g:X\times X\to Y\times Y$ is defined by $g(x_{1},x_{2})=(f(x_{1}),f(x_{2}))$. Then $f$ is uniformly continuous iff for any $V\in\mathcal{V}$, there is a $U\in\mathcal{U}$ such that $U\subseteq g^{{-1}}(V)$.

###### Proof.

Suppose $f$ is uniformly continuous. Pick any $V\in\mathcal{V}$. Then $U\in\mathcal{U}$ exists with $U[x]\subseteq f^{{-1}}(V[f(x)])$ for all $x\in X$. If $(a,b)\in U$, then $b\in U[a]\subseteq f^{{-1}}(V[f(a)])$, or $f(b)\subseteq V[f(a)]$, or $g(a,b)=(f(a),f(b))\in V$. The converse is straightforward. ∎

###### Proposition 2.

. If $f:X\to Y$ is uniformly continuous, then it is continuous under the uniform topologies of $X$ and $Y$.

###### Proof.

Let $A$ be open in $Y$ and set $B=f^{{-1}}(A)$. Pick any $x\in B$. Then $y=f(x)$ has a uniform neighborhood $V[y]\subseteq A$. By the uniform continuity of $f$, there is an entourage $U\in\mathcal{U}$ with $x\in U[x]\subseteq f^{{-1}}(V[y])\subseteq f^{{-1}}(A)=B$. ∎

Remark. The converse is not true, even in metric spaces.

## Mathematics Subject Classification

54E15*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections